A322084 - OEIS

%I #15 Jan 07 2024 15:52:15

%S 1,1,1,1,2,0,1,4,2,1,1,8,8,4,2,1,16,26,16,6,0,1,32,80,64,26,4,0,1,64,

%T 242,256,126,32,6,1,1,128,728,1024,626,208,48,8,1,1,256,2186,4096,

%U 3126,1280,342,64,7,2,1,512,6560,16384,15626,7744,2400,512,73,12,0,1,1024,19682,65536,78126,46592,16806,4096,703,104,10,0

%N Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n, n/d==1 (mod 4)} d^k - Sum_{d|n, n/d==3 (mod 4)} d^k.

%H Andrew Howroyd, <a href="/A322084/b322084.txt">Table of n, a(n) for n = 1..1275</a> (first 50 antidiagonals)

%H <a href="/index/Ge#Glaisher">Index entries for sequences mentioned by Glaisher</a>

%F G.f. of column k: Sum_{j>=1} j^k*x^j/(1 + x^(2*j)).

%e Square array begins:

%e   1,  1,   1,    1,     1,     1,  ...

%e   1,  2,   4,    8,    16,    32,  ...

%e   0,  2,   8,   26,    80,   242,  ...

%e   1,  4,  16,   64,   256,  1024,  ...

%e   2,  6,  26,  126,   626,  3126,  ...

%e   0,  4,  32,  208,  1280,  7744,  ...

%t Table[Function[k, SeriesCoefficient[Sum[j^k x^j/(1 + x^(2 j)), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten

%o (PARI) T(n,k)={sumdiv(n, d, if(d%2, (-1)^((d-1)/2)*(n/d)^k))}

%o for(n=1, 10, for(k=0, 8, print1(T(n, k), ", ")); print); \\ _Andrew Howroyd_, Nov 26 2018

%Y Columns k=0..12 give A002654, A050469, A050470, A050471, A050468, A321829, A321830, A321831, A321832, A321833, A321834, A321835, A321836.

%Y Cf. A109974, A279394, A279396, A285425, A322081, A322082, A322083.

%K nonn,tabl

%O 1,5

%A _Ilya Gutkovskiy_, Nov 26 2018